On Some Geometric Properties of Certain Köthe Sequence Spaces

It is proved that if a Köthe sequence space X is monotone complete and has the weakly convergent sequence coefficient WCS(X) > 1, then X is order continuous. It is shown that a weakly sequentially complete Köthe sequence space X is compactly locally uniformly rotund if and only if the norm in X is equi-absolutely continuous. The dual of the product space ( ⊕∞ i=1 Xi)Φ of a sequence of Banach spaces (Xi) ∞ i=1, which is built by using an Orlicz function Φ satisfying the ∆2-condition, is computed isometrically (i.e. the exact norm in the dual is calculated). It is also shown that for any Orlicz function Φ and any finite system X1, . . . , Xn of Banach spaces, we have WCS(( ⊕n i=1 Xi)Φ) = min{WCS(Xi) : i = 1, . . . , n} and that if Φ does not satisfy the ∆2-condition, then WCS(( ⊕∞ i=1 Xi)Φ) = 1 for any infinite sequence (Xi) of Banach spaces.